425,874 research outputs found
Dynamical Behaviour in the Nonlinear Rheology of Surfactant Solutions
Several surfactant molecules self-assemble in solution to form long, flexible
wormlike micelles which get entangled with each other, leading to viscoelastic
gel phases. We discuss our recent work on the rheology of such a gel formed in
the dilute aqueous solutions of a surfactant CTAT. In the linear rheology
regime, the storage modulus and loss modulus
have been measured over a wide frequency range. In
the nonlinear regime, the shear stress shows a plateau as a function
of the shear rate above a certain cutoff shear rate
. Under controlled shear rate conditions in the plateau regime,
the shear stress and the first normal stress difference show oscillatory
time-dependence. The analysis of the measured time series of shear stress and
normal stress has been done using several methods incorporating state space
reconstruction by embedding of time delay vectors.The analysis shows the
existence of a finite correlation dimension and a positive Lyapunov exponent,
unambiguously implying that the dynamics of the observed mechanical instability
can be described by that of a dynamical system with a strange attractor of
dimension varying from 2.4 to 2.9.Comment: 12 pages, includes 7 eps figure
Autoencoders for discovering manifold dimension and coordinates in data from complex dynamical systems
While many phenomena in physics and engineering are formally
high-dimensional, their long-time dynamics often live on a lower-dimensional
manifold. The present work introduces an autoencoder framework that combines
implicit regularization with internal linear layers and regularization
(weight decay) to automatically estimate the underlying dimensionality of a
data set, produce an orthogonal manifold coordinate system, and provide the
mapping functions between the ambient space and manifold space, allowing for
out-of-sample projections. We validate our framework's ability to estimate the
manifold dimension for a series of datasets from dynamical systems of varying
complexities and compare to other state-of-the-art estimators. We analyze the
training dynamics of the network to glean insight into the mechanism of
low-rank learning and find that collectively each of the implicit regularizing
layers compound the low-rank representation and even self-correct during
training. Analysis of gradient descent dynamics for this architecture in the
linear case reveals the role of the internal linear layers in leading to faster
decay of a "collective weight variable" incorporating all layers, and the role
of weight decay in breaking degeneracies and thus driving convergence along
directions in which no decay would occur in its absence. We show that this
framework can be naturally extended for applications of state-space modeling
and forecasting by generating a data-driven dynamic model of a spatiotemporally
chaotic partial differential equation using only the manifold coordinates.
Finally, we demonstrate that our framework is robust to hyperparameter choices
On dimension reduction in Gaussian filters
A priori dimension reduction is a widely adopted technique for reducing the
computational complexity of stationary inverse problems. In this setting, the
solution of an inverse problem is parameterized by a low-dimensional basis that
is often obtained from the truncated Karhunen-Loeve expansion of the prior
distribution. For high-dimensional inverse problems equipped with smoothing
priors, this technique can lead to drastic reductions in parameter dimension
and significant computational savings.
In this paper, we extend the concept of a priori dimension reduction to
non-stationary inverse problems, in which the goal is to sequentially infer the
state of a dynamical system. Our approach proceeds in an offline-online
fashion. We first identify a low-dimensional subspace in the state space before
solving the inverse problem (the offline phase), using either the method of
"snapshots" or regularized covariance estimation. Then this subspace is used to
reduce the computational complexity of various filtering algorithms - including
the Kalman filter, extended Kalman filter, and ensemble Kalman filter - within
a novel subspace-constrained Bayesian prediction-and-update procedure (the
online phase). We demonstrate the performance of our new dimension reduction
approach on various numerical examples. In some test cases, our approach
reduces the dimensionality of the original problem by orders of magnitude and
yields up to two orders of magnitude in computational savings
Equivalence of robust stabilization and robust performance via feedback
One approach to robust control for linear plants with structured uncertainty
as well as for linear parameter-varying (LPV) plants (where the controller has
on-line access to the varying plant parameters) is through
linear-fractional-transformation (LFT) models. Control issues to be addressed
by controller design in this formalism include robust stability and robust
performance. Here robust performance is defined as the achievement of a uniform
specified -gain tolerance for a disturbance-to-error map combined with
robust stability. By setting the disturbance and error channels equal to zero,
it is clear that any criterion for robust performance also produces a criterion
for robust stability. Counter-intuitively, as a consequence of the so-called
Main Loop Theorem, application of a result on robust stability to a feedback
configuration with an artificial full-block uncertainty operator added in
feedback connection between the error and disturbance signals produces a result
on robust performance. The main result here is that this
performance-to-stabilization reduction principle must be handled with care for
the case of dynamic feedback compensation: casual application of this principle
leads to the solution of a physically uninteresting problem, where the
controller is assumed to have access to the states in the artificially-added
feedback loop. Application of the principle using a known more refined
dynamic-control robust stability criterion, where the user is allowed to
specify controller partial-state dimensions, leads to correct
robust-performance results. These latter results involve rank conditions in
addition to Linear Matrix Inequality (LMI) conditions.Comment: 20 page
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